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"There's something in our world that makes men lose their heads—they couldn't be fair if they tried. In our courts, when it's a white man's word against a black man's, the white man always wins. They're ugly, but those are the facts of life. […]
"The one place where a man ought to get a square deal is in a courtroom, be he any color of the rainbow, but people have a way of carrying their resentments right into a jury box." (23.38-40)
According to Atticus, there's just something about race that makes white people crazy. His holding up Jem as an exception because of his youth suggests that whatever that X factor is, it's learned and not innate (and thus, perhaps can be changed through education?). He also acknowledges, in case it wasn't already obvious, that law isn't a pure realm free of the prejudices that plague everyday life—it's subject to the same problems as society at large. Usually Atticus is a voice of hope for change, but here he flatly says that racism is a "fact of life," suggesting that losing Tom's case severely dented his optimism concerning human nature—or else that, having sat through the case, Jem is ready to hear a truer, grimmer version of how the world works, instead of the sanitized Disney version.
[Atticus says] "As you grow older, you'll see white men cheat black men every day of your life, but let me tell you something and don't you forget it—whenever a white man does that to a black man, no matter who he is, how rich he is, or how fine a family he comes from, that white man is trash." Atticus was speaking so quietly his last word crashed on our ears. I looked up, and his face was vehement. "There's nothing more sickening to me than a low-grade white man who'll take advantage of a Negro's ignorance. Don't fool yourselves—it's all adding up and one of these days we're going to pay the bill for it. I hope it's not in you children's time." (23.40)
Here's a quick reminder that To Kill a Mockingbird wasn't written in the 1930s, when it takes place, but in the 1950s, in the middle of the sometimes violent civil rights movement: grown up Jem and Scout are paying for it.
"She has committed no crime, she has merely broken a rigid and time-honored code of our society, a code so severe that whoever breaks it is hounded from our midst as unfit to live with. She is the victim of cruel poverty and ignorance, but I cannot pity her: she is white. She knew full well the enormity of her offense, but because her desires were stronger than the code she was breaking, she persisted in breaking it." (20.43)
It's an enormous injustice to have Tom on trial and pre-convicted for something he didn't do. But Mayella is also a victim of injustice: dirt poor, kept ignorant, raped by her father, and forbidden to seek companionship from the one person who was ever nice to her. No surprise that Atticus is the one to see it.